ON AUTOMOPHISMS OF GROUP VON NEUMANN ALGEBRAS

1. INTRODUCTION

Group von Neumann algebras of discrete groups are an important source of examples of finite von Neumann algebras and several

authors ( [2], [6], [13], [15], [16]) have studied their natural *- automorphisms, i.e. those induced by characters and by automor- phisms of the group.

In this note we generalize some of the known results by ele- mentary methods and complement· the exposition given in [18, Sec- tions 22.10-22.13].

·we

first describe the group generated by natural *-automorphisms and give some criterions for properly

outerness. Secondly, we obtain some relations between fixed point algebras and crossed products of group von Neumann algebras which may be of interest in view of the isomorphism problem for such

algebras. The final example section is mainly devoted to non inner amenable groups, since the associated group factors are then known to be full ( [1], [10]) and thus· being far less understood.

vve now fix ,some notation. Hhen no reference or definition is given, the reader may consult [18] and/or some of the standard text books in the respective fields.

All groups will be considered as discrete groups and G will always denote such a non-trivial group, with identity element e.

·'

cl.(G) will denote the· (group) von Neumann algebra generated by the left regular representation g ~ A(g) of G on 1 2 (G). ~(G) is a (group) factor if and only if G is ICC, in which case it is a rr_1-factor.

An element ·AE£(G) is usually identified with fA= Ao E12(G), where

o

denotes the characteristic function of {e}, and we set supp(A)

=

{gEGifA(g):fO}. Then when H is a subgroup of G, we may identify

cl

(H) with {AE

J:(

G)

I

supp (A) :;: H}.

r will always denote the character group of G, with identity element 1, and CG the commutator subgroup of G.

Let H be a von Neumann algebra. When <!J is a group-homo- morphism from a group K into Aut(G) (resp. Aut(H)), the semi- direct product of G (resp. the crossed product of M) by the action ^<1> of K is denoted by G x<!JK (resp. H x<I>K).

At last, for nE{l,2, .•.

,oo},

~ will denote the cyclic group n

with n elements and

Jr

the free group on n generators.

n

Acknowledgements: Our hearthy thanks go to Erling St¢rmer for his support during this work.·

2. ON THE NATURAL *-AUTOMORPHISMS OF .C(G)

2 .1. For yEr (resp. crEAut(G)), we let a

y (resp. ~ )

cr denote the *-automorphism of dt(G) induced by y (resp. cr), and a

(resp. ~) the associated action of r (resp. Aut(G)) into

Aut(~(G)). We recall that

ay(A(g))

=

y(g)A(g), ~cr(A(g))

=

A(cr(g)) ^{( g} EG) .

2.2. He define N(c£..(G)) to be the subgroup of Aut(c£.(G)) gener- ated by a(r) U ~(Aut(G)). Let <!J:Aut(G) ~ Aut(r) be the action defined by

and define

<l>cr(y)

=

yocr -1 (crEAut(G), yEf),

i:f x<I>Aut(G) ~ N(~(G)) by i(y,cr)

=

a ~ •

y cr

Then we have:

Proposition: The mapping i is an isomorphism of r x¢Aut(G) onto N(eC.(G)).

Proof: One obtains immediately that

( 1) a: ^{Q - Q} a: ^Q a: -a: ^Q (yEr,· aEAut(G)).

yf'y- f'a yoa' f'a y - . yoa~1f'a

Thus

i((yl,al)(y2,a2))

=

i(yl¢a (y2),al a2) 1

= i(yl(y2oa~l),aia2)

=

^{a -1 ~}

Y1(Y2°a1 ) a1a2

=

^a: Y1 (y2oa1 ) a1 ^{a: -1 ~ ~} a2

=

^{a: ~ a: ~}

Y 1 ^{a l} Y 2 ^a2

=

i(yl,al)i(y2,a2)

for all y.Er, a.EAut(G), ~

=

1,2.

J J

Further, i is easily seen to be injective and it follows from (1)

that i is onto.

•

·'·

2.3. Some of the results of [2; Sections 4, 5] may be viewed as criterions for the outerness of elements of N(.(.(G)). On the other hand, recall that ( [13], [18; prop. 22.12]), for aEAut(G), we

have: ~a is properly outer (or freely acting) if and only if

( 2) the set {a(a)ga- 1 ;aEG} is infinite for every gEG.

These approaches may be unified as follows.

Lemma: Let

e =

a ~ , yEr, crEAut(G), and let AEl(G) be a

e-

Y cr

dependent element, i.e. AB

=

8(B)A, for all BE~(G).

Then we have:

i) y(cr(a))fA(g)

=

fA(cr(a)ga ^-1 ) (a,gEG),

ii) the set {cr(a)ga ^-1 ~aEG} is finite for every gEsupp (A),

iii) supp(A) lies in a coset of Go, where Go denotes the normal subgroup of G consisting of all elements in G with finite conjugacy classes.

Proof: By assumption we have: AA(a)

=

8(A(a))A (aEG), which gives y(cr(a))A

=

A(cr(a))~AA(a) ^~.

=

A(cr(a) ^-1 )Aida), from which i) follows. ii) follows from i) and the fact that JfAJE~² (G), while iii) is immediate from ii).

2.4. Proposition: Let 8EN(~(G)) be given as crEAut(G). Then

e

is properly outer whenever:

i) cr satifies (2) or

e =

^a ~ ^I yEr,

y cr

•

ii) y*l, cr is inner and at least one of the following conditions is satisfied:

a) Go agrees with the center

z

of G.

b) y is of infinite order.

·'·

c) the set {a'ga ^-1 ~aEG} is infinite for every gEGY

=

^{gEGJy(g)

=

^1}, g*e·

Proof: Let A be a e-dependent element in l(G). If i) holds

. -

then lemma 2.3 ii) implies that supp(A)

=

¢, i.e. A

=

0 and thus

e

is properly outer. Next, suppose y*l and cr is inner. Then

~cr is inner, and

e

may suppose

e =

^{a •} _y

will be properly outer if a is. Thus we

y

We will now apply lemma 2.3 i) and ii) (with cr

=

identity). Suppose A*O and let bEsupp(A). Here we obtain

that bEG 0 and that the centralizer of b in G is a subgroup of GY. Thus the index of Gy in G must be finite, i.e. y is of finite order, and further G

=

GY, i.e. y

=

1 if G0

= z.

So by contraposition,

e

is properly outer if a) or b) is satisfied. If now c) is satisfied, one easily obtains that GYn supp(A)

= ¢.

Since a _y (A)

=

A by [4], we have also supp(A) ~ Gy (cf. proposi- tion 3.1). Thus supp(A)

=

¢, i.e.

e

is properly outer.

•

The essence of [ 2 ~ Cor. 1 and 2, p. 589] is that cr is outer if and only if cr satisfies (2) when G is an R-group or has no normal subgroups of finite index other than itself. Observe also that G0

=

Z trivially when G is ICC or abelian and that c) is especially satisfied when Gy is ICC.

2.5. Let yEr. Since yocr

=

y whenever crEAut(G) is inner, i.e.

crEint(G), one may also consider r x~ Out(G), where

<l>

Out(G)

=

Aut(G)/Int(G) and <j>:Out(G) ~ Aut(r) is the action defined by:

~;(y)

=

<j>cr(y) (; denoting the coset of crEAut(G)).

Then we have:

Proposition: Suppose G0

= z.

Then

i) The action a of r in ~(G) is properly outer.

ii)

e =

^ay~cr (yEr, crEAut(G)) is inner if and only if y

=

1 and cr is inner.

iii) Out(~(G))

=

Aut(J(G))/Int(£(G)) contains a subgroup isomorphic to r X~ Out(G).

Proof: i) This follows from proposition 2.4 ii) a).

ii) Suppose 9 = Ad(U), U unitary in ~(G). Let bEsupp(U) and set cr' = ad(b)Eint(G). Then

~:7e

=

Ad(~(b)*u) EN(~(G))

and

*

^-1

~(b) U is a ~cr·9-dependent element in L(G) such that

*

e Esupp(~(b) U) since f~(b)*u(e) = fu(b)*O. By lemma 2.3 iii) we obtain:

supp(~(b)*u) ~

G0 = Z, i.e.

~::a

is the identity auto- morphism and so y = 1 and cr = cr'Eint(G) by proposition 2.1.

The converse is trivial.

iii) Let n:Aut(~(G)) + Out(L(G)) denote the canonical homomor- phism and define n' = noi where i:r x~Aut(G) +N(~(G)) is defined in 2.2. Then ker n' = {{1, cr) ;a Eint(G)} by ii), and one checks that (r x~Aut(G))/ker n' ~ r x¢ Out(G) under the obvious isomer- phism. Thus r x~ Out(G) ~ n' (r x~Aut(G)).

•

3. FIXED-POINT ALGEBRAS AND CROSSED PRODUCTS

3.1. Fixed-point algebras of J((G) under automorphisms induced by characters have a nice description:

Proposition: Let yEr, r• be a subgroup of r and set

·'·

Gy = {gEGjy(g)=l}, Gr' = n Gy and Then we have:

yEr'

0: 0:

i) J._(G) Y "',l(GY), where c£..(G) Y ii) _.[(G)o:

'

"'i,(G

r•

), where

= {A E ((G)

I

^0: y (A) =A} .

0:

=

n .l..(G) Y yEr'

iii) l(G)o: "'~(CG). Especially, a is ergodic if and only if G

iv)

is abelian.

N

=

^{GY, G}

r•

neither.

or CG is not inner amenable whenever G is

Proof:

i)

ii)

Let AE£(G). Then a (A) = A <=>

y

y(g)fA(g) = fA(g) (gEG) <=> fA(g) = 0 (gEG, g*GY).

a

Hence /_(G) y = {AE.((G) isupp(A) ~ GY} "'.[(GY).

a

n ~(G) Y = {AE{(G)

I

supp(A) c

yEr'

iii) We have that

r

CG = G and that CG

= {

e} if and only if is abelian ( cf.

[11.;

th. 23.8]), so iii) follows from ii).

iv) Since N contains CG, G/N is abelian and thus amenable.

The result now follows from [1; Cor. 2 iv)]~

G

•

Corollary: Suppose G is a countable ICC-group and let

r•

be a finite subgroup of r of order n. Set a' =air' and

r•

M = the

n

nxn-complex matrices. Then: £(G) X _a ^I

r

^{I "'} £,( G ) ® _, Mn.

Proof: Combine the proposition, proposition 2.5 i) and [6]. 8

3.2. Consider now an exact sequence of groups:

1 ~ H ~ G ~ K ~ 1.

When the extension splits, one may write G as a semidirct product of H by K and so ( cf. [ 18; 22. 10]) there exis,ts an action

<j!:K ~ Aut(c(.(H) such 'that

It follows from the deep [19; th. 6.1] that the same conclusion is true when £(H) is a II 1-algebra and K is finite. However, there exist extensions (cf. [9]) where such a conclusion is not possible and one is then forced to introduce a so-called regular extension of ,C,.(H) by K (see [7], [19] for definitions and other results).

Vile now show how cyclic extensions may be handled by elementary methods. From now on, we identify H with its image in G.

Proposition: Suppose the above extension is cyclic, i.e. K is cyclic. Then there exists an action ~:K ~ Aut(L(H)) such that:

Moreover, the action ~ is properly outer whenever H satisfies:

( 4) the set {hgh- 1 :hEH} is infinite for every gEG, g*H.

Proof: Suppose first K

=

^{;£. I} n<+oo.

n Then there exists an such that G is generated by a and H, and such that:

a m

¢H

(l~m~n-1), while a ⁿ EH.

Now, let V be an n-th root of 1-.(a ) ⁿ in o(.( H) and set aEG

*

~

*

U = V 1-.(a), which is an unitary in ~(G). Then ~(A) = UAU , (AEL(H)) defines an *-automorphism of ~(H), since H is normal in G, which is such that

~n(A) = (v*t-.(a))nA(t-.(a)*v)n

= ((v*)nt-.(an))A(t-.(an)*vn) =A (AE((H)),

·'·.

since V commutes with 1-.(a) .

*

So we may define an action

~:K ~ Aut(,C(H)) by

' 1•• (A) = ,,,j (A) = UjA(Uj )* (AE r(H) 'EK)

'I' 'I' - . . . , ] .

J

Clearly, ~(G) is generated by ~(H) and U. Further, let

E: ~{G) ~£(H) be the canonical conditional expectation, which is such that E(l-.(g)) = 0 when gEG, g*H. Then

(jEK, j:fO).

Thus the first part (3) follows from [18; 22.2] in this case. Hhen K = ~. the extention splits and (3) again follows. We may also clearly proceed along the same lines as above.

The second part may be verified by direct computation. It is also a consequence of [18; 22.3] since (4) is equivalent to

J:(H) ¹noL(G) '=.~(H). •

Corollary: Let yEf be of finite order n ^' ( resp. such that and let K¹ ,= 'l. (resp. Z).

n Then there exists an action

a a

<V ¹ : K ¹ ~ Aut (

.C

(G) Y) such that:

L

(G) "'

J..

(G) Y x ^{<V 1} K ^{1 ,} and which is

properly outer if Gy satisfies (4).

Proof: Combine the proposition and proposition 3.1.

•

3.3. If H satisfies:

( 5) the set {hgh ^-1 :hEH} is infinite for every gEG, g:fe, (i.e.

(H) ¹n (G) reduces to the scalars)

then H satisfies (4) and both H and G are ICC.

We have:

Proposition: For gEG, let I = {hEHihg=gh}.

g Suppose H is not

amenable (resp. not inner amenable) while I is amenable (resp.

g

inner amenable)·-for all gEG, g*H. Then H satisfies (4). If the same is true for all gEG, g:fe, then H satisfies (5).

Proof: Suppose that X

=

{ hg 0h -l ; hE H } is finite for some g 0 EG, go=l=e. Then consider the action of H on X defined by con juga-

~"'

tion. For xEX, [1; addendum],

the isotropy subgroup is then I

.

Therefore,

X

H is amenable (resp. inner amenable) if I is

X

amenable (resp. inner amenable) for all xEX. Since X ^c G'{e}

by

and X c G\H if g₀EG\H, the result follows by contraposition. •

3.4. Suppose that in 3.2, K ^{~ ~ x ~· •} Then we may apply propo- nl n2

sition 3.2 twice and obtain that there exist an action

'

~1 ^:Zl -+ Aut(.(.(H)) and an action ~2^::1 -+ Aut(.((H)x,1• ~ ⁾ such

nl n2 ~1 nl

that:

.(_(G)

This generalizes clearly to the case when K is abelian and finitely generated.

4. SOME EXAMPLES AND OPEN PROBLEMS

In each case we only give some relevant details. For all asser- tions about non inner amenability, we refer to [1]. Non inner amenable groups are automatically ICC.

4.1. Let G

= f =

<a 1 , ... ,a ^{>, 2~n~+oo.} Then G is not inner

n n

amenable and we have that r ~ Tn (T denoting the circle group) under the isomorphism given by y-+ (y(a 1 ), ..• ,y(an)).

Further G y ( resp. Gr' ) is a free subgroup of G whose ,rank depends on the order of G/Gy (resp. G/G

r•

) ( c f • [ 14 i th . 2 . 1 0

J ) •

Especially, Gy~ Wm(n-l)+l if y is of finite order m, while Gy~

W

otherwise. An easy application of proposition 3.3 shows

00

that Gy always satisfies condition (5}. Corollary 3.2 may now be applied specifically to obtain analoguous statements of known

results. For example, the case n

=

2, y

=

(A,A}, AET, is the one studied in [6], while the case ^y

=

exp(2ni/m,l, ... ,1} corresponds to [16; prop. 4.5]. At last, Aut(G} is described in [14; sect.

3.5]. The condition (2} is easily verified for many crEAut(G}.

4.2. Let G be the free product Then G is

not inner amenable when p (or q} >2, while G is not ICC when p

=

q

=

2. From [14; p. 193-197], we may obtain what follows.

First, we have that

CG"" f(p-l}(q-l} and G / CG "" Z. x 1 . p q Hence £.(G}a o:[('[r(p-l}(q-l}) (by 3.1} and

J:.(G}

""L<.t<r(p-l}(q-l}}x<V 1 :~-P}x<V 2 ^Zq

(by 3.4).

Let us be more specific for G

=

^~₂^*~₃o:PSL(2,~).

Here CG "' fF ₂ and r "' a ₂ ^x _~3 "" ^~6.

Thus .(_(~*Z3)a ""/...(F2)' cl.< a2*~ ^{> X a} ~ ^{"' .((IF 2) ® H6}

and .(<~*~) "" .( (F 2} x<V a6' where <V::l6 ^~ Aut (.[(lF 2 )) is the

action obtained in 3.2, this being outer by 3.3. At last, observe

·'·

4.3. As we have seen in 2.5, Out(L(G}) contains a copy of

r x~ Out(G) when G is ICC. It would be interesting to know if

(jl

these groups are-more intimately related, at least in some cases.

Connes has shown in [8] that Out(L(G)) is countable whenever .G is ICC and has property T (G is then especially non inner amenable}. We now mention two examples of such groups for which r x~ Out(G} "" ~2 ^. In both cases, it is·an open problem whether Out (

L (

G } ) "' ~ •

a) Let G

=

SL ( 3, ~) .

Here

r =

{1} while Out ( G ) "' Z by [ 12 ] .

2 A representative

~EAut(G) of the non trivial element in Out(G) is defined by

~(a)

=

(at)-l (aEG), where at denotes the transpose of a.

b) Let G

=

SL ( 3 , :;!.) x ~ ~ .

Now one may check that r "' ~ while Out(G) is trivial.

Observe also that ,/:..,(G) xa ~ "' .( ( SL ( 3, :l.)) ® M2 .

4.4. In the same spirit one may ask whether Out (l._( G) ) and

'

r ^x~

<P

Out(G) are isomorphic for some G. Another open problem whether r ^x~

<P

Out(G) (or even more Out(.{( G) ) ) may be trivial some countable non inner amenable group G. Besides other ICC groups with property T, we mention as possible candidates:

-the Ol'shanskii group ( [17])

-the amalgams of the type F 2

*

F 2 constructed in [3].

F 00

is for

These groups are simple and thus at least with trivial character group.

References:

1. E. Bedos, P. de laHarpe: Moyennabilite interieure des groupes:

definitions et exemples. Ens. Hath. (to appear).

2. H. Behncke: Automorphisms of crossed products.

Tohoku J. Math.

11: (

1969), 580-600.

3. R. Carom: Simple free products.

J. London Hath. Soc. 28 (1953), 66-76.

4. H. Choda: On freely acting automorphisms of operator algebras.

Kodai Hath. Sem. Rep. 26 (1974), 1-21.

5. H. Choda: A comment on the Galois theory of.finite factors.

Proc. Jap. Acad. 50 (1974)., 619-622.

6. M. Choda: Automorphisms of finite factors on free groups.

Math. Jap. 22 (1977), 219-226.

7. H. Choda: Some relations of

rr

₁-factors on free groups.

Math. Jap. 22 (1977),. 383-394.

8. A. Connes: A factor of type

rr

₁ with cotintable fundamental group. J. Operator theory 4 (1980}, 151-153.

9. A. Connes,

v.

Jones: Property T for von Neumann algebras.

Bull. London Math. Soc. 17 (1985}, 57-62.

10. E. G. Effros: Property r and inner amenability.

Proc. Amer. Math. Soc. 47 (1975}, 483-486.

11. E. Hewitt, K. Ross: Abstract Harmonic Analysis, vol. I.

Springer Verlag 1963.

12. L. K. Hua, I. Reiner: Automorphisms of the unimodoular group.

Trans. Amer. Math. Soc. 71 (1951), 331-348.

13. R. R. Kallmann: A generalization of free action.

Duke J. Hath. 36 (1969), 781-789.

14.

w.

Magnus, A. Karass, D. Solitar: Combinatorial group theory.

Interscience 1966.

15.

w.

L. Paschke: Inner product modules arising from compact

,1,.

automorphisms gro,ups of von Neumann algebras~·

Trans. Amer. math. soc. 224 (1976), 87-102.

16. J. Phillips: Automorphisms of full

rr

1 -factors with applica- tions to factors of type III.

Duke J. Math. 43 (1976), 375-385.

17. A. Yu. Ol'shanskii: On the problem of the existence of an invariant mean on a group.

Russian Math. Surveys 35 (1980}, 180-181.

18. S. Stratila: f1odular theory in operator. algebras.

Editira Academiei-Abacus Press 1981.

19. C. E. Sutherland: Cohomology and extensions of von Neumann algebras II. Publ. RIHS (Kyoto Univ.) 16 (1980}, 135-174.